Joint convergence of conditional expectations

Let P_n, nIN{0}, be probability measures on a-fieldA; f_n, nIN{0}, be a family of uniformly boundedA-measurable functions andA _n, nIN, be a sequence of sub--fields ofA, increasing or decreasing to the-fieldA _o. It is shown in this paper that the conditional expectations converge in P_o-measure to with k, n, m , if P_n|A, nIN, converges uniformly to P_n|A and f_n, nIN, converges in P_o-measure to f_o.     

Logarithms and imaginary powers of closed linear operators

The imaginary powersA ^it of a closed linear operatorA, with inverse, in a Banach spaceX are considered as aC ₀-group {exp(itlogA);t R} of bounded linear operators onX, with generatori logA. Here logA is defined as the closure of log(1+A) – log(1+A ^–1). LetA be a linearm-sectorial operator of typeS(tan ), 0(/2), in a Hilbert spaceX. That is, |Im(Au, u)| (tan )Re(Au, u) foru D(A). Then ±ilog(1+A) ism-accretive inX andilog(1+A) is the generator of aC ₀-group {(1+A) ^it ;t R} of bounded imaginary powers, satisfying the estimate (1+A) ^it exp(|t|),t R. In particular, ifA is invertible, then ±ilogA ism-accretive inX, where logA is exactly given by logA=log(1+A)–log(1+A ^–1), and {A ^it;t R} forms aC ₀-group onX, with the estimate A ^it exp(|t|),t R. This yields a slight improvement of the Heinz-Kato inequality.     

Reconstructing dimensions of intersections of convex sets

Denoting by dimA the dimension of the affine hull of the setA, we prove that if {K _i:i T} and {K _i ^j :i T} are two finite families of convex sets inR ⁿ and if dim {K _i :i S} = dim {K _i ^j :i S}for eachS T such that|S| n + 1 then dim {K _i :i T} = dim {K _i : {i T}}.     

Über Positive Resolventenwerte Positiver Operatoren

In this paper we study the positive resolvent values of positive operators respectively of positive elements in Banach lattice ordered algebras. In the matrix case these values are just the inverse M-matrices. One of the main results is the following: Let A be a Banach lattice ordered algebra. A positive invertible element xA is a resolvent value of a positive element yA if and only if the element x satisfies the negative principle: If aA, < 0 and xaa then xa 0.     

Kite-freeP- andQ-polynomial schemes

LetY = (X, {R _i } _oid) denote aP-polynomial association scheme. By a kite of lengthi (2 i d) inY, we mean a 4-tuplexyzu (x, y, z, u X) such that(x, y) R ₁,(x, z) R ₁,(y, z) R ₁,(u, y) R _i–1,(u, z) R _i–1,(u, x) R _i. Our main result in this paper is the following.     

The periodic Dirac operator is absolutely continuous

We consider the periodic Dirac operatorD inL ₂( ^d ). The magnetic potentialA and the electric potentialV are periodic. Ford=2 the absolute continuity ofD is established forA,VL _{r, loc} ,r>2; the proof is based on the estimates, obtained by the authors earlier [BSu2] for the periodic magnetic Schrödinger operatorM. Ford3 our considerations are based on the estimates forM, obtained in [So] forAC ^2d+3 . Under the same condition onA, forVC, the absolute continuity ofD, d3, is proved. ForA=0 the arguments of the paper give a new (and much simpler) proof of the main result of [D].The research was completed in the framework of the project INTAS-93-351.     

On the product of sign vectors and unit vectors

For a fixed unit vectora=(a ₁,...,a _n )S ^n-1, consider the 2 ⁿ sign vectors=(₁,..., _n ){±1{ ⁿ and the corresponding scalar products^·a = _n ⁱ⁼¹ = _i a _i . The question that we address is: for how many of the sign vectors must^.a lie between–1 and 1. Besides the straightforward interpretation in terms of the sums ±a ₂ , this question has appealing reformulations using the language of probability theory or of geometry.The natural conjectures are that at least 1/2 the sign vectors yield |.a|1 and at least 3/8 of the sign vectors yield |.a|<1 (the latter excluding the case when |a _i |=1 for somei). These conjectured lower bounds are easily seen to be the best possible. Here we prove a lower bound of 3/8 for both versions of the problem, thus completely solving the version with strict inequality. The main part of the proof is cast in a more general probabilistic framework: it establishes a sharp lower bound of 3/8 for the probability that |X+Y|<1, whereX andY are independent random variables, each having a symmetric distribution with variance 1/2.We also consider an asymptotic version of the question, wheren along a sequence of instances of the problem satisfying ||a||0. Our result, best expressed in probabilistic terms, is that the distribution of .a converges to the standard normal distribution, and in particular the fraction of sign vectors yielding .a between –1 and 1 tends to 68%.This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation.     

Ordinary and strong density continuity of complex analytic functions

In the paper we prove that the complex analytic functions are (ordinarily) density continuous. This stays in contrast with the fact that even such a simple function asG:²²,G(x,y)=(x,y ³ ), is not density continuous [1]. We will also characterize those analytic functions which are strongly density continuous at the given pointa . From this we conclude that a complex analytic functionf is strongly density continuous if and only iff(z)=a+bz, wherea, b andb is either real or imaginary.     

Equazioni ellittiche in forma di divergenza con 2∘ membro misura su domini non limitati

Summary Let a regular open set of R n, a measure with compact support and L a second order elliptic operator in divergence form. If L is coercive we prove a theorem of existence and uniqueness for the solution of Lu=, uH ₀ ¹+H₀ ^1,p()where p is the conjugate of p[n, ].     

Einige Aspekte in der Zuordnungstheorie

Zusammenfassung Gegeben seien endliche MengenX, Y undZ X × Y, Z _x ={y¦(x,y) Z},Z _y ={x¦(x,y) Z}.Man nenntA X (bzw.B Y)zuordenbar, wenn es eine Injektion:A Y (bzw.: B X) mit(x) Z _x (bzw.(y) Z _y ) gibt, und (A, B) mit #A=#B > 0 einZuordnungspaar, wenn eine Bijektionf:A B mitf(x)Z _x B (bzw.f ^–1 (y) Z _y A) existiert. Die Bijektionf heißtZuordnungsplan fürA, B.In der vorliegenden Arbeit werden Fragen nach der Existenz von optimal zuordenbaren Mengen und optimalen Zuordnungspaaren behandelt, wenn man auf den MengenX undY Ordnungen vorgibt, wobei auch Nebenbedingungen berücksichtigt werden. In manchen Fällen lassen sich anhand der Beweise Zuordnungspläne oder ihre Berechnungsvorschrift explizit angeben.Zum Schluß werden die Aussagen an konkreten, dem Bereich der Wirtschaftswissenschaften entnommenen Beispielen erläutert.

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Summary LetX, Y be finite sets andZ X × Y, Z _x ={y¦(x,y) Z},Z _y ={x¦(x,y)Z}. A X (resp.B Y) is calledassignable if there is an injection: A Y (resp.: B X) with (x) Z _x (resp.(y) Z _y ), (A, B) with #A=#B > 0 anassigned pair if there is a bijection f:A B withf (x) Z _x B (resp.f ^–1(y) Z _y A). The bijectionf is called aplan forA andB.In this paper problems are discussed concerning the existence of optimal assignable sets and optimal assigned pairs ifX andY are totally ordered, additional constraints are also considered. In some cases the proofs give explicit constructions of plans. The results are illustrated by application to problems occurring in Operations Research.

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Diese Arbeit ist mit Unterstützung des Sonderforschungsbereiches 72 an der Universität Bonn entstanden.     

On improved regularity of weak solutions of some degenerate,Anisotropic elliptic systems

Summary We consider a (possibly) vector-valued function u: R^N, Rⁿ, minimizing the integral , 2-2/(n*1)<p<2, whereD _i u=u/x _i or some more general functional retaining the same behaviour, we prove higher integrability for Du: D₁ u,..., D_n–1 u L^p/(p-1) and D_nu L²; this result allows us to get existence of second weak derivatives: D(D₁ u),...,D(D_n–1u)L² and D(D_n u) L _p.This work has been supported by MURST and GNAFA-CNR.     

On properties of the probabilistic constrained linear programming problem and its dual

In this paper, the two problems inf{inf{cx:x R ⁿ,A ₁ xy,A ₂ xb}:y suppF R ^m,F(y)p} and sup{inf{uy:y suppF R ^m,F(y)p}+vb:uA ₁+vA ₂=c, (u,v0} are investigated, whereA ₁,A ₂,b,c are given matrices and vectors of finite dimension,F is the joint probability distribution of the random variables ₁,..., _m, and 0<p<1. The first problem was introduced as the deterministic equivalent and the second problem was introduced as the dual of the probabilistic constrained linear programming problem inf{cx:P(A ₁ x)p,A ₂ xb}.b}. Properties of the sets and the functions involved in the two problems and regularity conditions of optimality are discussed.     

Polynomial functions over commutative semi-groups

Let S be a commutative semigroup. We consider the semigroup P(S) with respect to composition of all transformations p: S S of the form x a,x xⁿ or x axⁿ (a S; n N) and the semigroup P(S) containing only elements of the last two forms. Since all polynomials over S have the form a, xⁿ or ax these transformations are the so-called polynomial functions over S. We investigate the relationship between the structures of S and (S) resp. P(S) — a criterion on the commutativity of S has been shown by means of polynomial functions in 2.     

Waterman classes and triangular sums of double Fourier series

Let ^a ={nln^a (n+1)}, where a R. The following results are established: For every &fnof ^a BV ((- ]²), the triangular partial sums of its Fourier series are uniformly bounded if a = -1, and converge everywhere if a < -1.For every a>0, there exists &fnof ^a BV ((- ]²) such that the triangular partial sums of its Fourier series are unbounded at the point (0;0).     

Topology of Definable Hausdorff Limits

Let A ^n+r be a set definable in an o-minimal expansion S of the real field, let A ^r be its projection, and assume that the non-empty fibers A_a ⁿ are compact for all a A and uniformly bounded, i.e. all fibers are contained in a ball of fixed radius B(0,R). If L is the Hausdorff limit of a sequence of fibers A_ai, we give an upper-bound for the Betti numbers b_k(L) in terms of definable sets explicitly constructed from a fiber A_a. In particular, this allows us to establish effective complexity bounds in the semialgebraic case and in the Pfaffian case. In the Pfaffian setting, Gabrielov introduced the relative closure to construct the o-minimal structure S_Pfaff generated by Pfaffian functions in a way that is adapted to complexity problems. Our results can be used to estimate the Betti numbers of a relative closure (X,Y)₀ in the special case where Y=.     

On a classical Nicoletti boundary value problem

We shall derive existence, uniqueness and comparison results for the functional differential equationx(t)=f(t, x), a. e.tI, with classical Nicoletti boundary conditionsx _i(t_i)=y _iX, iA, whereI is a real interval,A is a nonempty set andX is a Banach space.     

Finite rank,relatively bounded perturbations of semigroups generators

Summary This paper is motivated by, and ultimately directed to, boundary feedback partial differential equations of both parabolic and hyperbolic type, defined on a bounded domain. It is written, however, in abstract form. It centers on the (feedback) operator A_F=A+P; A the infinitesimal generator of a s.c. semigroup on H; P an Abounded, one dimensional range operator (typically nondissipative), so that P=(A·, a)b, for a, b H. While Part I studied the question of generation of a s.c. semigroup on H by A_F and lack thereof, the present Part II focuses on the following topics: (i) spectrum assignment of A_F, given A and a H, via a suitable vector b H; alternatively, given A, via a suitable pair of vectors a, b H; (ii) spectrality of A_F—and lack thereof—when A is assumed spectral (constructive counterexamples include the case where P is bounded but the eigenvalues of A have zero gap, as well as the case where P is genuinely Abounded). The main result gives a set of sufficient conditions on the eigenvalues {_n} of A, the given vector a H and a given suitable sequence {_n} of nonzero complex numbers, which guarantee the existence of a suitable vector b H such that A_F possesses the following two desirable properties: (i) the eigenvalues of A_F are precisely equal to _n+_n; (ii) the corresponding eigenvectors of A_F form a Riesz basis (a fortiori, A_F is spectral). While finitely many _ns can be preassigned arbitrarily, it must be however that _n 0 « sufficiently fast ». Applications include various types of boundary feedback stabilization problems for both parabolic and hyperbolic partial differential equations. An illustration to the damped wave equation is also included.Research partially supported by Air Force Office of Scientific Research under Grant AFOSR-84-0365.     

Regularity and Unit-regularity of Generalized Semigroups of Linear Transformations

Let V and W be vector spaces over a division ring D and L_D (V, W) the set of all linear transformations from V into W. For L_D(W, V), let (L_D (V, W), ) denote the semigroup L_D (V, W) with the operation * defined by * = for all , L_D(V, W). By a unit-regular semigroup we mean a semigroup S with identity having the property that for each a S, a = aua for some unit u S. The main purpose of this paper is to prove the following statements. The semigroup (L_D(V, W), ) is regular if and only if V = {0}, W = {0} or is an isomorphism from W onto V. The semigroup (L_D (V, W), ) is unit-regular if and only if (i) V = {0}, (ii) W = {0} or (iii) is an isomorphism from W onto V and dim_D V .     

Spectral properties of cosine operator functions

LetA be the generator of a cosine functionC _t ,t R in a Banach spaceX; we shall connect the existence and uniqueness of aT-periodic mild solution of the equationu = Au + f with the spectral property 1 (C _T ) and, in caseX is a Hilbert space, also with spectral properties ofA. This research was supported in part by DAAD, West Germany.     

Connected and alternating vectors: Polyhedra and algorithms

Given a graphG = (V, E), leta ^S, S L, be the edge set incidence vectors of its nontrivial connected subgraphs.The extreme points of = {x R ^E: a^sx |V(S)| - |S|, S L} are shown to be integer 0/± 1 and characterized. They are the alternating vectorsb ^k, k K, ofG. WhenG is a tree, the extreme points ofB 0,b ^kx 1,k K} are shown to be the connected vectors ofG together with the origin. For the four LP's associated with andA, good algorithms are given and total dual integrality of andA proven.On leave from Swiss Federal Institute of Technology, Zurich.